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consistent radial fall
Patxi Ritter, S. Aoudia, Alessandro D.A.M. Spallicci, Stéphane Cordier
To cite this version:
arXiv:1511.04277v1 [gr-qc] 13 Nov 2015
Indirect (source-free) integration method. II. Self-force consistent radial fall
Patxi Ritter1,2,3, Sofiane Aoudia4,5, Alessandro D.A.M. Spallicci1,6∗, St´ephane Cordier2,7 1Universit´e d’Orl´eans
Observatoire des Sciences de l’Univers en r´egion Centre, UMS 3116 Centre Nationale de la Recherche Scientifique
Laboratoire de Physique et Chimie de l’Environnement et de l’Espace, UMR 7328 3A Av. de la Recherche Scientifique, 45071 Orl´eans, France
2Universit´e d’Orl´eans
Centre Nationale de la Recherche Scientifique
Math´ematiques - Analyse, Probabilit´es, Mod`elisation - Orl´eans, UMR 7349 Rue de Chartres, 45067 Orl´eans, France
3Univerzita Karlova
Matematicko-fyzik´aln´ı fakulta, ´Ustav teoretick´e fyziky V Hole˘sovi˘ck´ach 2, 180 00 Praha 8, ˘Cesk´a Republika
4Laboratoire de Physique Th´eorique - Facult´e des Sciences Exactes
Universit´e de Bejaia, 06000 Bejaia, Algeria
5Max Planck Institut f¨ur Gravitationphysik, A. Einstein
1 Am M¨uhlenberg, 14476 Golm, Deutschland
6Chaire Fran¸caise, Universidade do Estado do Rio de Janeiro
Instituto de F´ısica, Departamento de F´ısica Te´orica
Rua S˜ao Francisco Xavier 524, Maracan˜a, 20550-900 Rio de Janeiro, Brasil
7Universit´e Joseph Fourier
Agence pour les Math´ematiques en Interaction Centre Nationale de la Recherche Scientifique
Laboratoire Jean Kuntzmann, UMR 5224
Campus de Saint Martin d’H´eres, Tour IRMA, 51 rue des Math´ematiques, 38041 Grenoble, France (Dated: 14 September 2015)
We apply our method of indirect integration, described in Part I, at fourth order, to the radial fall affected by the self-force. The Mode-Sum regularisation is performed in the Regge-Wheeler gauge using the equivalence with the harmonic gauge for this orbit. We consider also the motion subjected to a self-consistent and iterative correction determined by the self-force through osculating stretches of geodesics. The convergence of the results confirms the validity of the integration method. This work complements and justifies the analysis and the results appeared in Int. J. Geom. Meth. Mod. Phys., 11, 1450090 (2014).
PACS numbers: 02.60.Cb, 02.60.Lj, 02.70.Bf, 04.25.Nx, 04.30.-w, 04.70Bw, 95.30.Sf
Keywords: Modelling of wave equation, General relativity, Equations of motion, Two-body problem, Gravi-tational waves, Self-force, Black holes.
Mathematics Subject Classification 2010: 35Q75, 35L05, 65M70, 70F05, 83C10, 83C35, 83C57
I. INTRODUCTION AND MOTIVATIONS
We apply the indirect method of Part I to the motion of a particle perturbed by the back-action, that is the influence of the emitted radiation and of the mass m0on its own worldline, thanks to the interaction with the field of the other mass M .
The problem of the back-action for massive point particles moving in a strong field with any velocity has been tackled by concurring approaches all yielding the same result, exclusively defined in the harmonic (H) gauge. Result derived in 1997 by Mino, Sasaki and Tanaka [1], Quinn and Wald [2], around an expansion of the mass ratio m0/M . The main achievement has been the identification of the regular and singular perturbation components, and their playing or not-playing role in the motion, respectively. The conclusive equation has been baptised MiSaTaQuWa from the first two initials of its discoverers. Later, Detweiler and Whiting [3] have shown an alternative approach, not any longer based on the computation of the tails, but derived from the Dirac solution [4]. It is customary to call self-force (SF) the expression resulting from MiSaTaQuWa and DeWh approaches, and to switch between the former (the SF externally breaking the background geodesic as non-null right hand-side term) and the latter (the particle following a geodesic of the total metric, background plus perturbations) intepretations of the same phenomenon.
∗
A full introduction to the SF is to be found in [5], while for a first acquaintance the reader might be satisfied by the arguments exposed in [6].
In the MiSaTaQuWa conception, the gravitational waves are partly radiated to infinity (the instantaneous, also named direct, component), and partly scattered back by the black hole potential, thus forming back-waves (the tail part) which impinge on the particle and give origin to the SF. Alternatively, the same phenomenon is described by an interaction particle-black hole generating on one hand a field which behaves as outgoing radiation in the wave-zone, and thereby extracts energy from the particle; on the other hand, in the near zone, the field acts on the particle and determines the SF which impedes the particle to move on the geodesic of the background metric. From these works, it emerges the splitting between the instantaneous and tail components of the perturbations, the latter acting on the motion. Unfortunately the tail component can’t be computed directly, if not as a difference between the total and the instantaneous components. Instead, the DeWh approach reproduces the Dirac definition. It consists of half of the difference between the retarded and the advanced fields, to which is added an ad hoc field including the contributions from the past light inner cone, while avoiding non-causal future contributions.
The SF computation is not an easy task because the field perturbation is divergent at the position of the particle, and it is therefore necessary to use a suitable procedure of regularisation. The latter deals with the divergences coming from the infinitesimal size of the particle. In the Regge-Wheeler (RW) gauge [7], we benefit of the wave-equation and of the gauge invariance of its wave-function. Regrettably, the singularity in the perturbed metric has a complicated structure which has made impossible so far to find a suitable regularisation scheme. Nevertheless, current investigations attempt to identify gauge transformations between the RW and H gauges, e.g., Hopper and Evans [8, 9], or use numerical integration approaches that deal de facto only with the homogeneous form of the Regge-Wheeler-Zerilli (RWZ) equation [7, 10].
In the H gauge, a regularisation recipe in spherical harmonics and named Mode-Sum, was conceived by Barack and Ori [11, 12]. Such a procedure is to be carried out partially or totally in the H gauge. There is an exception though for a purely radial orbit. In this case, there is a regular connection between the RW and H gauges; thus the quantification of the SF may be carried out entirely in the RW gauge. Further, the outcome is invariant for these two gauges and all regularly related gauges [12]. Herein, we thus proceed with a detailed computation in toto in the RW gauge, announced by Barack and Lousto in [13] but never appeared.
In the ’70s, Zerilli computed the gravitational radiation emitted during the radial fall [10, 14, 15] into a Schwarzschild-Droste (SD) black hole [16–18]. Many studies followed later on. The first was from Davis et al. [19], who considered, in the frequency domain, the radiation emitted by a particle initially at rest in free fall from infinity. Later, Ferrari and Ruffini [20] resume, still in the frequency domain, the same system, but conferring an initial speed to the particle from infinity. The first to solve the problem of the fall of the particle still initially at rest but for a finite distance from the black hole were Lousto and Price in a series of papers [21–23], where they detail and give a numerical technique to deal with the point source in the time domain. Martel and Poisson [24] resume the same problem by proposing a family of parametrised initial conditions, all of them being solutions of the Hamiltonian constraint; further they study the influence of these initial conditions on the wave-forms and energy spectra.
Thirty years later, back-action - without orbital evolution - was partially analysed only in two works [13, 25], and with contrasting predictions (in the former Lousto suggests that back-action is repulsive for most modes, conversely to the latter where Barack and Lousto attribute always an attractive feature). We have largely commented these papers in [26]. Needless to say, the time shortness of the fall forbids any important accumulation of back-action effects but, from the epistemological point of view, radial fall for gravitation remains the most classical problem of all, and raising the most delicate technical questions. Early gravitational SF computation were carried out in the H gauge by Barack and Sago for circular [27, 28] and eccentric orbits [29].
In the context of the Extreme Mass Ratio Inspiral (EMRI) gravitational wave sources, the gravitational SF heavily impacts the wave-forms. It has been suggested to evolve the most relativistic orbits through the iterative application of the SF on the particle worldline, i.e., the self-consistent approach by Gralla and Wald [30, 31]. We implement it for the least adiabatic orbit of all, that is radial infall, using our integration method. The strict self-consistency would imply that the applied SF at some instant is what arises from the actual field at that same instant. So far this has been done only for a scalar charged particle around an SD black hole by Diener et al. [32], and never for a massive particle. For quasi-circular and inspiral orbits, dealt by Warburton et al. [33], Lackeos and Burko [34], the applied SF is what would have resulted if the particle were moving along the geodesic that only instantaneously matches the true orbit. Herein, we adopt the latter acception.
We thus study how the back-action affects the motion, the radiated energy and the wave-forms of a particle without and with the self-consistent approach. According to the different inclinations of the reader, his interest may raise from one or more of the following considerations.
Technical assessments and advancements
into account (without and with orbital evolution) allow to test our numerical integration scheme, as we expect similar results, while possibly appreciating any difference, among these three cases.
• The inclusion of back-action effects demands a sophisticated algorithm of at least fourth order, since considering third time derivatives of the wave-function.
• The contrasting results in [13, 25] need a resolution. We have recovered the results in [13] (self-force is attractive in H and RW gauges) and proved wrong those in [25] (claiming that the self-force is mainly repulsive and divergent at the horizon), see the full discussion in [26]. Anyway, the work in [13] does not consider the impact of the self-force on the trajectory, which we deal with herein.
• Radial infall is the least adiabatic orbit of all types. Imposing the identity between the radiated energy and the lost orbital energy for computing the corrections on the motion would be most unjustified as shown by Quinn and Wald [35]. Indeed, it is just for non-adiabatic orbits, that is required applying a continuous correction on the trajectory due to the SF effects, i.e. the self-consistent method [30, 31]. Thus, radial infall imposes such an application, though it is not rewarding due to the feebleness of the SF effects themselves.
• Given the limitations of numerical relativity in evolving circular and elliptic orbits for small mass ratio binaries, the comparison of results for head-on collisions from numerical and perturbation methods is of interest. • In particle physics, when referring to the transplanckian regime and black hole production, back-action has a
pivotal role in head-on collisions according to Gal’tsov et al. [36, 37].
• The regular transformation between H and RW gauges for radial trajectories allow to carry out the Mode-Sum regularisation entirely in the RW gauge.
When endeavouring towards astrophysical scenarios, we recall that
• It has been estimated by Amaro-Seoane, Sopuerta et al. [38, 39] that a relevant number of EMRIs will consist of direct plunges when the supermassive black hole (SMBH) is not rotating.
• Radial trajectories are comparable to portions of highly eccentric orbits producing an EMRB (Extreme Mass Ratio Burst) following Berg and Gair [40–42].
• The last stages of EMRI plunges were analysed by Keden, Gair and Kamionkowski [43] for discriminating supermassive black holes from boson stars, supposedly horizonless objects, and by Macedo et al. [44] for signatures of dark matter.
• The concept of maximal velocity in radial fall is discussed in high energy astrophysics for jets and tidal disruption by Chicone and Mashhoon [45, 46], Kojima and Takami [47].
General motivations
• Radial fall is the most classic problem in physics instantiated by the stone of Aristot´el¯es, the tower of Galilei, the apple of Newton, and the cabin of Einstein. The solutions represent the level of understanding of gravitation at a given epoch, and have thereby an epistemological relevance.
• It is a worthwhile problem `a la Feynman: The worthwhile problems are the ones you can really solve or help solve, the ones you can really contribute something to. No problem is too small or too trivial if we can really do something about it [48].
The paper is structured as follows. Section II, after a brief review of the SF, is largely devoted to the computation in the RW gauge of the regularisation parameters through the Mode-Sum method. Section III deals with some numerical issues, the performance and validation of the code. In Sect. IV, we deal with the impact of the SF on the motion of the particle without and with the self-consistent evolution for the radial fall through osculating orbits. The appendixes deal with the Riemann-Hurwitz regularisation [49, 50], the numerical extraction of the field at the particle position and display the jump conditions for the radial orbit.
II. GRAVITATIONAL SF A. Foreword
The SF equation, defined in the H gauge, is given by [1–3]
Fselfα = − 1 2m0(g αβ+ uαuβ)2hR(H) µβ;ν − h R(H) µν;β uµuν , (1)
where R stands for the regular part of the perturbations hµν, either tail (MiSaTaQuWa) or radiative (DeWh). The two contributions are not equivalent, but the final results are. The other quantities are the background metric gµν and the four-velocity uα. The SF is obtained by subtracting the singular part from the retarded force
Fselfα(H)= Fretα(H)− F α(H)
S . (2)
The retarded force is computed from the retarded field
Fretα(H)= Fα h hret(H)αβ i= m0kαβγδ∇δh ret(H) βγ = − 1 2m0 g αβ+ uαuβ 2∇δhret(H)βγ − ∇αhret(H)γδ uγuδ , (3) where hαβ= hαβ− 1/2gαβh, and kαβγδis given by
kαβγδ=1 2g αδuβuγ − gαβuγuδ−12uαuβuγuδ+1 4u αgβγuδ+1 4g αδgβγ . (4)
As shown in [12], for a transformation to any gauge (G)
hret(G)αβ = hret(H)αβ + δh(H→G)αβ , (5)
the SF changes as
Fselfα(G)=Fselfα(H)+ δFα(H→G)= Fretα(H)− F α(H) S + δFα(H→G)= F α(G) ret − F α(H) S . (6)
Thus, in an arbitrary gauge G, the singular term - to be extracted from the retarded force - is always expressed in the H gauge and not in the G one, as it might be supposed. In the H gauge, the isotropy of the singularity around the particle eases the computation of FSα(H), while guaranteeing its inconsequential role on the motion. Instead, in other gauges we are confronted with the lack of isotropy [2]. We recall the expression of the Mode-Sum decomposition in the H gauge [11] Fselfα(H)= ∞ X ℓ=0 Fretαℓ(H)− F α(H) S = ∞ X ℓ=0 Fretαℓ(H)− ∞ X ℓ=0 h Aα(H)L + Bα(H)+ Cα(H)L−1i− Dα, (7)
where L = ℓ + 1/2, and ℓ is the mode index. Inserting the Mode-Sum expression of FSα(H) from Eq. (7) into Eq. (6), and decomposing Fretα(G) in ℓ modes, we get [12]
Fselfα(G)= ∞ X ℓ=0 h Fretαℓ(G)− Aα(H)L − Bα(H)− Cα(H)L−1 i − Dα . (8)
Fselfα(G) = ∞ X ℓ=0 h Fretαℓ(G)− Aα(G)L − Bα(G)− Cα(G)L−1] − Dα(G) . (9)
In [12] several orbits were examined. It was concluded that the RW gauge is regularly connected to the H gauge only for purely radial orbits. Further, it has been shown that the components of the transformation gauge vector are not only regular at the position of the particle but they can be made vanishing. That is to say, for radial orbits, the regularisation parameters share the same expression in the RW and H gauges. The SF is thus gauge invariant for RW, H and all other gauges interrelated via a regular transformation gauge vector.
We thus derive the regularisation parameters entirely in the RW gauge, and confirm their identity with those in the H gauge found by Barack et al. [51]. The RW gauge has the distinct advantage of giving easy access to the components of the perturbation tensor (instead strongly coupled in the H gauge) via the RWZ wave-functions.
We deal from here onwards with radial infall. This implies that i) the odd modes vanish, and the source term for even modes is simplified; ii) there are not m modes; iii) the perturbation K vanishes for a fixed θ; iv) for symmetry, the terms Ft
self and Fselfr don’t vanish, conversely to Fselfθ = F φ self = 0.
B. Computation of the regularisation parameters in the RW gauge
The value x′
p(τ ) = (t′p, rp′) represents any point of the γ world-line followed by the particle, while xp = (t, rp) = x′
p(τ = 0) the point where the SF is evaluated, Fig. (1), τ being the proper time. Further, x = (t, r) indicates a point taken in the neighbourhood of xp where the field ψ(x) is evaluated, before taking the limit x → xp.
x b b b x′ p(τ ) xp(τ = 0) γ FIG. 1: We define x′ p(τ ) = (t ′ p, r ′
p) any point of the γ world-line followed by the particle, and xp= (t, rp) = x′p(τ = 0) the point
where we finally evaluate the SF, τ being the proper time; x = (t, r) indicates a point taken in the neighbourhood of xpwhere
the field ψ(x) is evaluated, before taking the limit x → xp. We define the Green function G (x, xp(τ )) as
ψ(x) = Z 0+ −∞ G x, x′ p(τ ) dτ . (10)
When associated to the RWZ equation [7, 10], we get h
− ∂t2+ ∂r2∗− V (r)
i
G = bG(r)δ(r − r′
p)δ(t − t′p) + bF(r)∂rδ(r − r′p)δ(t − t′p) , (11) where the even potential is
Veℓ(r) = 2f
λ2(λ + 1)r3+ 3λ2M r2+ 9λM2r + 9M3
where λ = (ℓ − 1)(ℓ + 2)/2, and the source term coefficients are b F(r) = − κrf 2(r) 4(λ + 1)(λr + 3M ) , b G(r) = 2(λ + 1)(λr + 3M )κrf r(λ + 1) − M 2r − 3M E2 λr + 3M , (13) for κ = 8πm0Yℓ0= 4m0 √ 2πL, and E = f(rp)ut.
According to the properties on distributions, Appendix (B) in Part I, Eq. (11) is rewritten using an alternative version of the Green function, named bG and defined such that
G(x, x′ p) = h b Q(r′ p) − bF(r′p)∂/∂rp′ i b G(x, x′ p) , (14) where bQ(r′ p) ··= h b G(r) − d bF(r)/dri r=r′ p . Using δ(r − r′ p) = f (r′p)−1δ(r∗− r′∗p) and Z ··= −∂t2+ ∂r2∗− V (r), Eq. (11) becomes ZG(x, x′ p) = " b G(r) − d bF(r)dr # r=r′ p δ(r − r′ p)δ(t − t′p) − bF(r′p) ∂ ∂r′ pδ(r − r ′ p)δ(t − t′p) = b Q(r′p) − bF(r′p) ∂ ∂r′ p δ(r − r′p)δ(t − t′p) = Xδ(r − r′p)δ(t − t′p) (15) ZX bG = XZ bG = Xδ(r − rp′)δ(t − t′p) , (16) h − ∂2 t + ∂r2∗− V (r) i b G = f (r′ p)−1δ(r∗− r′∗p)δ(t − t′p) . (17) Considering the causal structure of the Green function, it is now useful to introduce the Eddington-Finkelstein coordinates (u, v) [52, 53]. In these new variables, v = t + r∗ ingoing and u = t − r∗ outgoing, the expression of the wave-operator is simply given by ∂2
r∗− ∂t2= −4∂uv. In the same way, casting δ(r∗− r′∗p)δ(t − t′p) in (u, v) variables,
requires to deal with the product of a function µ(t, r∗) with δ. Under the integral definition of the latter Z µ(R2) δ(x)φ(x)dx = Z R2δ(µ(x))φ(µ(x)) |J µ| dx , (18) where φ ∈ D(R2) and |J
µ| is the determinant of the Jacobian matrix associated to µ |Jµ| = 2. Then, in (u, v) coordinates, Eq. (17) turns into
h 4∂uv+ V (r) i b G = 2f (r′ p)−1δ(u − u′p)δ(v − v′p) . (19)
The Mode-Sum regularisation in the RW gauge, and thus the determination of the SF, will be achieved by the un-dertaking of two pursuits (i) the analytic computation of the regularisation parameter; (ii) the numerical computation of the ℓ-modes of the retarded force by solving the RWZ equation.
For the analytic venture, the regularisation of the SF by the Mode-Sum technique requires the evaluation of the divergency, i.e. the singular part Fα
S of the retarded solution. The singular part is fitted by a 1/L power series, of which coefficients are the regularisation parameters. The computation of the latter is based on a local analysis, i.e. at the neighbourhood of the particle, of the wave-function ψ, or more exactly of its associated Green’s function. The technique consists of a perturbative expansion of the Green function modes in a small spacetime region around the particle for great values of ℓ.
causal nature of its support, i.e., its non-zero value in the future light cone of x′
p. Then, we expand the reduced Green function in powers of 1/L, where each term of the series, namely Gn, will also be locally expanded for x → xp, such that the coefficients of the expansion Gnwill be function of xp and x′p.
The next step considers integrating G with respect to the proper time τ . To this end, we have to express the Gn coefficients explicitly in terms of τ . This is achieved by a Taylor series of Gn x′p(τ )
around τ = 0. Finally, we integrate G and ∂rG to get the asymptotic behaviour of ψℓ and ∂rψℓ when ℓ → ∞. The computation of the other derivatives ∂n
t∂mr G, is performed by borrowing from Part I the relationships on partial derivatives, which were used for the jump conditions to get ∂n
t∂rmψℓ→∞ quantities for which n + m ≤ 3.
We will gain access to the behaviour of the wave-function and its derivatives versus ℓ, thereby testing the convergence of our numerical code for very high modes. We will then compute the ℓ-modes of the perturbations Hℓ
1,2, K and of the retarded force Fαℓ[hretℓ
αβ], for large values of ℓ, thereby accomplishing the other (numerical) venture.
The reader may skip this very technical discussion and get directly to the results expressed by Eqs. (120). Otherwise, we assume the reader be well acquainted with the Mode-Sum by Barack [54, 55] and the coming after literature.
1. Computation strategy and detailed description
For the asymptotic behaviour of ∂n
t∂rmψℓ when ℓ → ∞ for n + m ≤ 3, that is up the third derivative of the wave-function, we apply our strategy through the following steps
• a. Reduced Green’s function.
• b. Expansion of the reduced Green function in powers of 1/L around x = xp. • c. Reconstruction of the Green function G(x, x′
p). • d. Expansion around x′
p= xp. • e. Computation of ψℓ→∞(x
p), ∂rψℓ→∞(xp), and ∂tn∂rmψℓ→∞(xp).
a. Reduced Green’s function. The causal structure allows to rewrite bG in a reduced form g(x, x′
p). Indeed, bG(x, x′p) has support in the future light cone of x′
p, so bG(u < u′p, v) = bG(u, v < vp′) = 0. Thus
b
G(x, x′p) = 2f (r′p)−1g(x, x′p)H(u − u′p)H(v − vp′) , (20) where H(u − u′
p)H(v − vp′) are Heaviside or step distributions, which confine the support of bG(x, x′p) to the area made by all points x belonging to the future light cone of x′
p. By inserting Eq. (20) into Eq. (19), we express the wave-operator applied to bG ∂uv h g(x, x′ p)H(u − u′p)H(v − v′p) i = ∂u h ∂vgHu′ pHvp′ + gHu′pδv′p i = ∂uvgHu′ pHv′p+ ∂vgδu′pHvp′ + ∂ugHu′pδv′p+ gδu′pδv′p . (21) Equation (19) involves four distinct types of quantities
• (i) (· · · ) × Hu′ pHv ′ p, • (ii) (· · · ) × δu′ pHv ′ p, • (iii) (· · · ) × Hu′ pδv′p, • (iv) (· · · ) × δu′ pδvp′.
The action of each term relies upon the behaviour along the characteristic lines u = u′
p and v = vp′. • (i). For u > u′
pand v > v′p, only the term (i) has a contribution; g satisfies the homogeneous equation associated to Eq. (19).
• (ii). If u = u′
pis constant, only the term (ii) has a contribution; then ∂vg(u′p, v) = 0. • (ii). If v = v′
• (iv). On the world line (u, v) = (u′
p, v′p), the coefficient of term (iv) must be equal to the coefficient of the source term of Eq. (19); then g(u′
p, vp′) = 1. According to (i), we have
4∂uvg + V (r)g = 0 , ∀ u > u′p and v > vp′ , (22) while according to (ii), (iii) and (iv), we have
g(u = u′p, v) = g(u, v = vp′) = 1 . (23)
Equation (23) is in fact the initial condition to be associated with Eq. (22); it ensures the uniqueness of the solution. Figure (2) shows the support of the reduced Green function.
FIG. 2: The reduced Green function bG(x, x′
p) has support in the future light cone of x ′
p(dark grey area); thus bG(u < u ′ p, v) = 0
(blue area with top-right oblique lines) and bG(u, v < v′
p) = 0 (pink area with top-left oblique lines).
b. Expansion of the reduced Green function in powers of 1/L around x = xp. We are now looking for a solution g of Eqs. (22,23) near the evaluation point x = xp, that is r = rpwhile considering large values of ℓ. Thus, we Taylor expand the quantities around r = rp, and express them as power series in 1/L. For dealing with both very small quantities such as the spatial separation r − rp and large quantities proportional to L, we introduce new variables of the product form L × small spatial separation; these variables are called ”neutral” by Barack [54]. We first consider this procedure for the potential. In the neighbourhood of r = rp, or similarly around r∗= rp∗, we have
and ∆r∗
p = r∗− rp∗. The asymptotic behaviour of V(0), V(1) and V(2) for 1/L → 0 is
V(0)(rp) = f r2 p L2− 6M rp + 1 4 + O L−1= V(2)(0)L2+ V(0)(0)+ O L−1 , (26) V(1)(rp) = f r2 p " 6M − 2rp r2 p L2− 96M 2− 33Mr − r2 p 2r3 p !# + O L−1= V(2)(1)L2+ V(0)(1)+ O L−1 , (27) V(2)(r p) = f r2 p " 60M2− 40Mr p+ 6r2p 2r4 p L2−1152M 3− 810M2r p+ 124M rp2+ 3r3p 4r5 p # +O L−1 = V(2) (2)L 2+V(2) (0)+O L −1 . (28) In this notation, V(n)(k) refers to the n-th Taylor coefficient in 1/L of the k-th coefficient in ∆r∗
p. Figure (3) shows the geometric representation of the neutral variables.
x b b b x′ p(τ ) xp γ ∆r∗ p Y X Z u= u′ p v = vp′
FIG. 3: Geometric representation of the neutral variables used in the local analysis of g(x, x′
p). The grey area shows the support
of the Green function corresponding to the set of points belonging to x, the chronological future of x′p. The expanded potential becomes
V (rp) = V(0)(rp) + V(2)(1)(rp)L2∆r∗p | {z } O(L) + V(0)(1)(rp)∆r∗p | {z } O(L−1) + V(2)(2)(rp)L2∆r∗2p | {z } O(1) + V(0)(2)(rp)∆rp∗2 | {z } O(L−2) +O ∆rp∗3 , (29)
where we labelled the order of each term. The terms such as Ln∆r∗n
p n ∈ N are of 0th order, and do not catch the behaviour of V with respect to L because O(L−1) ∼ O(∆r∗
p). We then choose to introduce the neutral variables (underlined) for which the product form can be appraised as constant. We define
∆r∗p ··= L∆r∗p . (30)
with ν1= 2 rp 3M rp − 1 , ν2= −6M rp − 1 4 , ν3= 1 r2 p 3 − 20Mr p + 30M2 r2 p . (32)
Similarly to Eq. (30), we introduce X and Y as two neutral variables such that X ··= 12ρ(rp)L(u − u′p) and Y ··=
1
2ρ(rp)L(v − v ′
p) , (33)
where the pre-factor ρ(rp) ··= f(rp)1/2/rp simplifies Eq. (22) after the change of variables
∂uvg = (1/4)ρ(rp)2L2∂XYg (34)
is made. In addition, we introduce another neutral variable Z Z ··= 2pXY = ρ(rp)L
q (u − u′
p)(v − vp′) = Z = (L/rp)s , (35)
where s is the geodesic distance between the point x and the point x′
p. Indeed, in the SD metric, we have ds2= −fdudv, and therefore s =Rxx′ p gαβdxαdxβ 1/2
≈qf (rp)(u − u′p)(v − vp′). We define also the variable ∆r′∗p as
∆r′∗p ··= ρ(rp)L(r′∗p − r∗p) . (36)
The equation to be solved is now
∂XYg + h 1 + ν1∆rp∗ L−1+ ν2+ ν3∆r∗2p L−2+ O L−3 ig = 0 , (37)
where the reduced Green function g is to be expressed as a power series of 1/L, whose coefficients are function of ∆r∗p, ∆r′∗p and Z only g = ∞ X k=0 L−kgk(∆r∗p, ∆r′∗p, Z) . (38)
However, from a practical point of view, to get the desired accuracy, it is sufficient to truncate the sum at k = 2. Equation (22) becomes 2 X k=0 L−k∂XYgk+ h 1 + (f1∆r)L−1+ (f2+ f3∆2r)L−2+ O(L−3) iX2 k=0 L−kgk = 0 . (39)
Now, by identifying powers of L, we will have a hierarchical system of equations supplemented by the initial conditions, Eq. (23), of the form
∂XYgk+ gk= Sk ,
gk(u = u′p, v) = gk(u, v = v′p) = δk0 .
Concretely, we have
order 0 : ∂XYg0+ g0= 0 , (41)
order 1 : ∂XYg1+ g1= −ν1∆r∗pg0 , (42)
order 2 : ∂XYg2+ g2= −ν1∆r∗pg1− (ν2+ ν3∆r∗2p )g0. (43) Through a a change of variable Z = 2√XY, the left hand side changes into
∂2 ∂X∂Ygk+ gk= Z 2∂2gk ∂Z2 + Z ∂gk ∂Z + Z 2g k . (44)
Thus, Eq. (41) implies to solve a Bessel equation of order 0 Z2∂ 2g 0 ∂Z2 + Z ∂g0 ∂Z + (Z 2 − 02)g0= 0 , (45)
which solution of is a Bessel function of the first kind of order 0
g0= J0(Z) . (46)
Equations (42,43) are also Bessel equations with source terms. By working on the relationships between neutral variables ∆r∗p, ∆r′∗p, X and Y, we can rewrite the source terms Sk solely as function of Z and of the difference Y− X = ∆r∗
p− ∆r′∗p. Implementing the relationships in Tab. I, the solutions of the Eqs. (42,43) are built compatibly with the initial conditions of Eq. (40)
g1= − 1 4ν1ZJ1(Z)(∆r ∗ p+ ∆r′∗p) , (47) g2= − 1 6ZJ1(Z) h ν3 ∆r∗2p + ∆r∗p∆r′∗p + ∆r′∗2p + ν2 i + 1 96Z 2J 2(Z) h 3ν21 ∆r∗p+ ∆r′∗p 2 − 8ν3 i + 1 96ν 2 1Z3J3(Z) . (48) S Solution of ∂XYg + g = S 0 J0(Z) J0(Z) ZJ1(Z)/2 (Y − X)J0(Z) (Y − X)ZJ1(Z)/4 (Y− X)2J 0(Z) Z2J2(Z) + 2(Y − X)2ZJ1(Z) /12 ZJ1(Z) Z2J2(Z)/4 (Y− X)ZJ1(Z) (Y − X)Z2J2(Z)/6 (Y − X)2ZJ1(Z) Z3J3(Z) + 3(Y − X)2Z2J2(Z) /24
TABLE I: Provision for the solution of the generalised Bessel equation ∂XYg + g = S with a source term S written itself with
a Bessel function [54].
c. Reconstruction of the Green function G(x, x′
p). Given the local behaviour of g for large modes g = g0+ g1L−1+ g2L−2+ O L−3
, (49)
we can reconstruct the function bG(x, x′
p) linked to g through
b G(x, x′
and itself connected to the Green function, Eqs. (10,11) through Eq. (14), recalled herein G = b Q(r′ p) − bF(rp′) d dr′ p b G , (51) with b F(r′p) = −κ h f2(r′p)L−4+ O L−6 i , (52) b Q(r′p) = κρ(r′p)2 L−2+ 9 4 − 4M r′ p L−4+ O L−6 . (53)
According to Eq. (51), the determination of G implies the derivative of bG with respect to r′
p. This term necessarily involves the derivatives of ∆r′∗p, Z, and Hu′
pHv′p, listed here below
d∆r′∗ p dr′ p = dr ′∗ p dr′ p d dr′∗ p ∆r′∗p = Lρ(rp)f (r′p)−1 , (54) dZ dr′ p = Lρ(rp)f (rp′)−1 d dr′∗ p q (u − up)(v − vp) = Lρ(rp)f (rp′)−1 Y − X Z = Lρ(rp)f (r′p)−1 ∆r∗p− ∆r′∗p Z ! , (55) d dr′ p h H(u′− up)H(v − v′p) i = f (r′p)−1 h Hv′ pδ(u − u ′ p) − Hu′ pδ(v − v ′ p) i . (56)
Equation (56) involves two non-vanishing terms of which the contribution depends on how the evaluation point is reached, from the right r → r+
p or from the left r → r−p, Fig. (4). Therefore, for a simpler notation we adopt two additional neutral variables, displayed with the others in Tab. II
FIG. 4: The value of the derivative, Eq. (56), depends on the limit, either taken on the right r → r+
p or on the left hand-side
r → r−p of the evaluation point. If r → r −
p - left panel - the term involving δ(v − v ′
p) in Eq. (56) is null; instead, if r → r+p
-right panel - the term involving δ(u − u′
Variable Expression ∆r∗ p L(r ∗ −r∗ p) ∆r′∗ p ρ(rp)L(rp′∗−r ∗ p) X 1/2ρ(rp)L(u − u′p) Y 1/2ρ(rp)L(v − vp′) Z ρ(rp)L p (u − u′ p)(v − v′p) ω+ 2X ω− 2Y τ −Lτ
TABLE II: List of standard and neutral variables used and their expressions as function of L.
The above change of variable gives
δ(ω+) = δ(u − u′ p) dω + du −1 u=u′ p = L−1ρ(rp)−1δ(u − u′p) , (58) and δ(ω−) = δ(v − v′p) dω − dv −1 v=v′ p = L−1ρ(rp)−1δ(v − vp′) . (59) Therefore, d dr′ p h Hu′ pHv′p i = ±Lρ(rp)f (r′p)−1δ(ω±) . (60)
From Eqs. (20,49,53), the first term of G = bQ(r′
p) bG − bF(r′p)d bG/drp′ transforms into b Q(r′ p) bG =2 bQ(r′p)f (r′p)−1g(x, x′p)H(u − u′p)H(v − vp′) = κf (rp) 2rp L−2+ 9 4 − 4M rp L−4 + O(L−6) 2f−1(rp) h g0+ g1L−1+ g2L−2+ O(L−3) i θupθvp =κ r′ p g0L−2+ g1L−3+ g2+ g0 9 4− 4M r′ p L−4+ O L−5 Hu′ pHv ′ p . (61)
In Eq. (64) the derivative of Z, computed with Eq. (55), and of Jn(Z) are used. Useful properties are d dZ h ZnJn(Z) i = ZnJn−1(Z) , J−1(Z) = −J1(Z) . (66)
By multiplying Eq. (62) by Eq. (52), we get
b Fd bG1= κ4M r′2 p J0(Z)L−4+ O L−5 , (67) b Fd bG2= κρ(rp) ∆r∗p− ∆r′∗p J1(Z) Z L −3+1 2κν1ρ(rp) ∆r 2∗ p − ∆r′2∗p J0(Z)L−4+ 1 2κν1ρ(rp)ZJ1(Z)L −4 + O L−5 , (68) b Fd bG±3 = ∓2κρ(rp)J0(Z)δ(ω±)L−3∓ 2κρ(rp)g1δ(ω±)L−4+ O L−5 . (69) We inject Eqs. (61,67-69) into Eq. (14) to get G(x, x′
p) in power series of 1/L
G = G0L−2+ G±1L−3+ G±2L−4+ O(L−5) , (70)
where the coefficients Gn depend on r through ∆r∗p, and on r′p through ∆r′∗p G0= κ r′ p J0(Z) , (71) G±1 = κ rp h rp r′ p g1− 2f(rp)1/2 ∆r∗p− ∆r′∗p J1(Z) Z ± 2J0(Z)δ(ω ±)i, (72) G± 2 = κ rp ( rp r′ p g2+ " 1 4 rp r′ p 9 −32Mr′ p −12ν1f (rp)1/2 ∆r2∗p − ∆r′2∗p # J0(Z)− 1 2ν1f (rp) 1/2ZJ 1(Z) ± 2g1δ(ω±) ) . (73) d. Expansion around x′
p= xp. The behaviour of the ψ wave-function for large values of ℓ is achieved by integrating Eq. (70) over the world line. According to Eq. (10), the integration of G, in proper time τ , imposes first rendering the coefficients Gn(x = xp) explicitly function of τ ; put otherwise, expanding Gnin powers of τ around the evaluation point x′
p(τ ) = xp. We proceed as follows
• The evaluation of Eq. (70) at x = xp for r → rp and ∆r∗p → 0. Then, all Gn coefficients will be only function of r′
p and rp. • All r′
p-dependent quantities are expanded in powers of τ around the point r′p= rp, that is to say around τ = 0 up to order τ2. This will lead us to introduce a neutral time variable τ ∝ Lτ.
• t constant τ , we find the expansion of G in powers of 1/L such that
G = eG0(τ )L−2+ eG±1(τ )L−3+ eG±2(τ )L−4+ O(L−5) , (74) where the coefficients eGn(τ ) explicitly depend upon τ and L through τ .
• The integration of G to determine ψℓ→∞(t, r
p) involves terms proportional to τkJn(τ ), with k, n ∈ N. Caution is to be exercised, improper integrals arise.
• The whole procedure is applicable to ∂rG to get ∂rψℓ→∞(t, rp). So, first we take Eqs. (71-73), while requiring that r → rpand ∆r∗p→ 0
G0= κ r′ p
G±1 = κ rp rp r′ p g1− 2f(rp)1/2∆r′∗p J1(Z) Z ± 2J0(Z)δ(ω ±) , (76) G±2 = κ rp ( rp r′ p g2+ " 1 4 rp r′ p 9 − 32Mr′ p −12ν1f (rp)1/2∆r′2∗p # J0(Z)− 1 2ν1f (rp) 1/2ZJ 1(Z) ± 2g1δ(ω±) ) , (77) with g1= − 1 4f1ZJ1(Z)∆r ′∗ p , g2= − 1 6ZJ1 f3∆r′∗2p + 3ν2+ 1 96Z 2J 2(Z)3ν12∆r′∗2p − 8f3+ 1 96f 2 1Z3J3(Z) , (78)
wherein all quantities ω±, Z, ∆r′∗
p are taken at r = rp. The quantities depending upon rp′, and consequently on τ , in Eqs. (75-77) are ∆r′∗
p, 1/r′p, Z and Jn(Z), named collectively Q. Thus, an expansion around r′
p= rp corresponds to an expansion around τ = 0, since rp = r′p(τ = 0). Let Γ(τ ) be one of the quantities Q, then the Taylor expansion can be written as
Γ = Γ(τ = 0) + dΓ dτ τ =0 τ +1 2 d2Γ dτ2 τ =0 τ2+1 6 d3Γ dτ3 τ =0 τ3+ · · · , (79)
where τ is a small entity (τ → 0) such that O (τ) ∼ O (1/L). Introducing a neutral time variable
τ =−Lτ , (80)
for constant τ we obtain the following expansion
Γ = Γ0+ Γ1(τ )L−1+ Γ2(τ )L−2+ Γ3(τ )L−3+ · · · , (81) where the coefficients Γn(τ ) depend on τ and L only through τ . The coefficients Γn(τ ) are shown in Tab. (III) for each quantity Q. Γ Γ0(τ ) Γ1(τ ) Γ2(τ ) ∆r′∗p −f(rp)1/2˙rp∗τ 1 2f (rp) 1/2r pr¨′∗pτ2 − 1 6f (rp) 1/2r2 p˙¨rp′∗τ3 1 r′ p 1 rp ˙r′ p rp τ ˙r ′2 p rp − ¨ r′ p 2τ2 Z − ˙sτ 12rpsτ¨ 2 − 1 6r 2 p˙¨sτ3 Jn(Z) Jn(τ ) 1 2rps¨ dJn(τ ) dτ τ 2 −16rp2˙¨s dJn(τ ) dτ τ 3+1 8r 2 ps¨2 d2J n(τ ) dτ2 τ 4
TABLE III: Taylor coefficients Γ = Γ0+ Γ1L−1+ Γ2L−2+ O L−3when Γ is one of the Q quantities.
˙r∗ p =f (rp)−1˙rp , ¨ r∗ p =f (rp)−2 f (rp)¨rp− f′(rp) ˙r2p , ˙¨r∗ p =f (rp)−3 h 2f′(rp)2− f′′(rp)f (rp)˙r3p− f′(rp)f (rp) ˙rpr¨p+ f (rp)2˙¨rp i , (82)
where the primes indicate derivation with respect to r, while the point to τ . Through the expression ds2 dτ2 = −f(r ′ p) du′ p dτ dv′ p dτ , (83)
we are led to the normalisation relation ˙up˙vp= f (rp)−1. Taking then successive derivatives with respect to τ at point r′
p= rp, we get ¨up˙vp+ ˙upv¨p = f′(rp)−1˙rp and ˙up˙¨vp+ 2¨up¨vp+ ˙¨up˙vp= f′′(rp)−1˙r2p+ f′(rp)−1¨rp. The following step is the derivation of s with respect to τ at the evaluation point τ = 0
˙s(τ = 0) = − 1 , ¨ s(τ = 0) =1 2f ′(r p)f−1(rp) ˙rp , ˙¨s(τ = 0) = 1 16f (rp)2 h (8f′′(r p)f (rp) − 13f′(rp)2) ˙rp2+ 8f′(rp)f (rp)¨r i +1 4f (rp)¨upv¨p. (84)
Finally, the derivatives of Jn(τ ) with respect to τ for n ≥ 0 dJn(τ ) dτ = Jn−1(τ ) − n τJn(τ ) = n τJn(τ ) − Jn+1(τ ) , (85) d2J n(τ ) dτ2 = 1 22 h Jn−2(τ )− 2Jn(τ ) + Jn+2(τ ) i = Jn n(n + 1) τ2 − 1 +Jn+1 τ . (86) e. Computation of ψℓ→∞(x
p), ∂rψℓ→∞(xp), and ∂tn∂rmψℓ→∞(xp). Introducing expansions of Q, Tab. (III), in Eqs. (75-77), we finally express the coefficients of G in function of the proper time
e G±2(τ ) = κ 96rp n −12h64M − 18rp− 4f(rp) −2M + 2f(rp)rp+ f (rp)1/2ν1rp ˙r∗2p τ2 + r2ps¨ f (rp)1/2ν1˙rp∗˙s + rps¨ τ4iJ0(τ ) + rpτ h 48ν2˙s − 24f(rp)3/2ν1˙rp∗2˙sτ2+ 16f (rp) ˙r∗p(ν3˙r∗p˙s − 3rps)τ¨ 2 + 4rpτ 4rp˙¨sτ ∓ 24δ(ω±)¨s + 3rp¨s2τ + 12f (rp)1/2ν1 4 ˙s ∓ 4δ(ω±) ˙r∗p˙sτ + rp(¨r∗p˙s + ˙rp∗(1 + ˙s)¨s)τ2 J1(τ ) + −8ν3˙s2+ 48f (rp)rp(¨rp∗+ 2 ˙rp∗¨s) + 3f (rp)ν12˙r∗2p ˙s2τ2 τ J2(τ ) − ν12˙s3τ2J3(τ ) io . (90)
Thus, eG±2 is built from the contributions coming from the terms of G±2 in O(τ0), of G±1 in O(τ1), and of G0 in O(τ2). All quantities else than τ , involved in the formulation of eG
n(τ ), are evaluated at rp. Returning to the definition given in Eq. (10), we can compute the integral of G with respect to τ
ψℓ→∞(x) = Z 0+ −∞ G x, xp(τ ) dτ = rp L Z +∞ 0− e G x, xp(ˆτ ) dˆτ =rp L Z +∞ 0− e G0(ˆτ )L−2+ eG1(ˆτ )L−3+ eG2(ˆτ )L−4 +O L−5 dˆτ = rp Z +∞ 0− e G0(ˆτ )L−3dˆτ | {z } ψOL3 + rp Z +∞ 0− e G1(ˆτ )L−4dˆτ | {z } ψ±OL4 +O L−5 . (91)
Integrals in Eq. (91) involve terms of the form Z +∞
0−
τkJ
n(τ )dτ k, n ∈ N , (92)
which diverge for certain values of k and n. Indeed, Jn(τ ) has an asymptotic behaviour of the form Jn(τ ) ≈ p
2/πτ cos(τ − nπ/2 − π/4). Thus, for large positive values of τ , the integrand will be of the formp2/πτmcos(τ − nπ/2 − π/4) with m = k − 1/2. To get a finite value from Eq. (92), we cancel the divergence through recasting the integral as [54] Z +∞ 0− → Z +f∞ 0− . (93)
The definition of the ”tilde” integral is given by the limit of the same name, i.e. the ”tilde limit” Z +f∞ 0− ··= f lim λ→+∞ Z λ 0− , (94)
where the limit, applied to any quantity K depending on λ, is given by f lim λ→+∞K(λ) = limλ→+∞ K(λ) −X j Oj(λ) . (95)
propose an example where 0 ≤ k ≤ n to show how the tilde limit acts concretely on the quantity to be regularised. Consider Ik
n(λ) the primitive of the function λkJn(λ)
Ink(λ) ··= Z
λkJn(λ)dλ . (96)
The integrand can be rewritten as
λkJn(λ) = λk−n−1 h λn+1Jn(λ) i = λk−n−1 d dλ h λn+1Jn+1(λ) i . (97)
Then, integration by parts leads to a recurrence relation on Ik n(λ) Ink(λ) = λkJn+1(λ) − (k − n − 1)In+1k−1(λ) = k−1 X j=0 (n − k − 1 + 2j)!! (n − k − 1)!! λ k−jJ n+1+j(λ) +(n + k − 1)!! (n − k − 1)!!I 0 n+k(λ) . (98)
Thus, using the tilde limit, the sum in Eq. (98) disappears because each term is of the form Oj(λ) when λ → +∞. The term proportional to I0
n+k(λ) is trivial since the standard integral R+∞
0 Jn(λ)dλ = 1 ∀n ≥ 0 is well defined and is finite. Accordingly,
Z +f∞ 0
λkJn(λ)dλ =(n + k − 1)!!
(n − k − 1)!! for 0 ≤ k ≤ n . (99)
Following the same reasoning, the general expressions depending on the values of the integers k and n are
Z +f∞ 0 τkJn(τ )dτ = (n + k − 1)!!/(n − k − 1)!!, if 0 ≤ k ≤ n, (−1)(k−n)/2(n + k − 1)!!/(n − k − 1)!!, if k − n > 0 is even, 0, if k − n > 0 is odd. (100)
Returning to the computation of ψℓ→∞, the evaluation of the integral in Eq. (91) is done by replacing the standard by a tilde integral. The first term O L−3is obvious and gives
ψOL3= rp Z +f∞ 0− κ rp J0(τ )dτ = κ . (101)
The term O L−4is written as ψOL4± = Z +∞ 0− κ 4 h ± 8δ(ω±)J 0(τ )− ν1f (rp)1/2˙rp∗˙sτ2J1(τ )− 2¨sτ2J1(τ )− 4f(rp) ˙r∗pτ J2(τ ) i dτ , (102) with δ(ω±) = δ(τ ) |dω±/dτ |−1 = |f ˙ω
∓| δ(τ ), the second equality is found by using the normalisation condition. Applying Eqs. (100), the integral is simplified and can be computed
ψ± OL4= ± 2κ Z +f∞ 0− |f ˙ω∓| δ(τ )J0(τ )dτ − κf(rp ) ˙r∗ p Z +f∞ 0− τ J2(τ )dτ = ±2κ |f ˙ω∓|τ =0J0(0) − κf ˙r∗p 2!! 0!! = − 2κfh˙rp∗∓ ˙ω∓ i = ±2κf ˙t = ±2κE . (103)
Finally, we obtain the formulation of ψ± on the world line for large modes ℓ → ∞
ψ±ℓ→∞= κhL−3± 2EL−4+ O L−5 i . (104)
∂rψ±ℓ→∞= κ rpf (rp) −1 " ∓ EL−2−32E2L−3± 6M rp − 9 4 EL−4+ O L−5 # . (105)
The next derivatives of ψ± are given by
∂rψ±= κ rp f (rp)−1 ∓EL−2−32E2L−3± 6M rp − 9 4 EL−4+ O(L−5) , (106a) ∂r2ψ±= κ r2 p f (rp)−2 h E2L−1± 2 −3Mr p EL−2+ O(L−3)i, (106b) ∂3 rψ±= κ r3 p f (rp)−3 ∓E3+ E2 5 2E 2+9M rp − 6 L−1∓ 3E 7M rp (M rp − 1) + 2 L−2+ O(L−3) , (106c) ∂tψ± = κ rp ± ˙rpL−2+ 3 2E ˙rpL −3 ∓ 6M rp − 9 4 ˙rpL−4+ O(L−5) , (106d) ∂t2ψ±= κ r2 p h E2− f(rp)L−1∓ E 2rp L−2+ O L−3 i, (106e) ∂t3ψ±= κ r3 pf (rp) ∓E E2− f(rp) +E 2˙r p 2rp h 16M + 5rp(E2− 1) i L−1±E ˙rr2p p 3M2− 2Mrp L−2+ O L−3 , (106f) ∂r∂tψ±= κ r2 pf (rp) h − E ˙rpL−1± 3M rp − 1 ˙rpL−2+ O(L−3) i , (106g) ∂r∂t2ψ±= κ r3 pf (rp) ∓E E2− f(rp) + 1 2r2 p h 5E4− 9E2+ 4r2p+ 24M E2− 20M rp+ 6 i L−1 ±rE2 p M rp− 3M2L−2+ O L−3 , (106h) ∂2 r∂tψ± = κ r3 p f (rp)−2 ±E2˙r p− E ˙rp 5 2E 2+9M rp − 4 L−1± ˙r p h 3M rp (5M rp − 4) + 2 i L−2+ O(L−3) , (106i)
and the asymptotic behaviour of the metric perturbation functions with respect to ℓ are given by
Equations (107,110) confirm that also for ℓ → ∞, the perturbations are continuous at the position of the particle, see Sect. IV. The perturbations K, although not used in the computation of the SF for the radial fall
K±= κ 1 2rp L−1+ O(L−2) , (113) ∂tK = κ ˙rp 2f (rp)rp2 ±E − E 2 2 L −1 + O(L−2) , (114) ∂rK±= κ 2f (rp)r2p ∓EE 2 2 − f L−1+ O(L−2) , (115) ∂trK±= κ ˙rp 2f (rp)2rp3 ( − E2L ± Eh5M − 2r pE(1 − E) i − E 2 2f rp 17M + 4rpE2− 11rp L−1+ O(L−2) ) . (116)
We recall the relation between the hret
αβ modes and the perturbation functions H1ℓ and H2ℓ
hretℓαβ = f Hℓ 2 H1ℓ Hℓ 1 f−1H2ℓ r 2ℓ + 1 4π . (117)
By putting Eqs. (107-112) in the expression of the retarded SF, Eq. (3), we get
Fretαℓ= − m0 2f " f0α ∂Hℓ 2 ∂t − df drH ℓ 1 + f1α ∂Hℓ 1 ∂t − df drH ℓ 2 + f2α ∂Hℓ 2 ∂r + f α 3 ∂Hℓ 1 ∂r # Yℓ0 . (118)
Recalling now the definition of the ℓ independent regularisation parameters
lim x→xp
Fsing±αℓ = Fret±αℓ→∞(rp) = A±αL + Bα+ CαL−1+ O(L−2) , (119) we obtain their explicit expression by equating each L power of the right and left-hand sides of Eq.(119). The regularisation parameters for a radial geodesic in an SD black-hole in the RW gauge are given by
Ar± = ∓m 2 0 r2 p E , At± = ∓ m 2 0˙r2p r2 pf (rp) , Br= −m20 2r2 p E2, Bt= − m20˙rp 2r2 pf (rp)E , Cα= 0 . (120)
where ˙t = E/f(rp) and ˙rp= p
E2− f(r
p). The parameters are to be put into Eq.(9), noting that Dα= 0 [13, 51].
C. Non-radiative modes
In absence of a wave-equation for the non-radiative modes ℓ = 0, 1, it is necessary to identify an alternative way for evaluating their contribution. Incidentally, the contributions of the radiative and non-radiative modes, though they refer to different gauges have been summed in previous literature [13].
1. Zerilli gauge
see also Detweiler and Poisson [56]. For the ℓ = 0 mode, it is possible to obtain an analytic solution for Fαℓ=0 ret . With the gauge transformation xα→ xα+ ξα
ℓ=0, we get ξαℓ=0= M0(t, r), M1(t, r), 0, 0 Y00 . (121)
The perturbations transform as
h(G’)αβ → h(G)αβ + ∇αξ ℓ=0
β + ∇βξαℓ=0 , (122)
and thus the components are related to the new gauge (G′) by, see Gleiser et al. [57] H0ℓ=0(G’)= H ℓ=0(G) 0 + 2 ∂ ∂tM (G→G’) 0 + 2M r2fM (G→G’) 1 , (123) H1ℓ=0(G’)=H1ℓ=0(G)− f−1∂t∂M1(G→G’)+ f ∂ ∂rM (G→G’) 1 , (124) H2ℓ=0(G’)= H ℓ=0(G) 2 − 2 ∂ ∂rM (G→G’) 1 + 2M r2fM (G→G’) 1 , (125) Kℓ=0(G’)= Hℓ=0(G)−2rM1(G→G’) . (126)
The Zerilli (Z) gauge [10] implies that the two degrees of gauge freedom M0 and M1 must render H1ℓ=0(Z) = Kℓ=0(Z)= 0. Frettℓ=0= m0 4f2 h 2f′f2˙r p˙t3p H2ℓ=0(Z)+H ℓ=0(Z) 0 −f3˙t4p+f ˙r2p˙t2−f2˙t2p+ ˙r2p ∂ ∂tH ℓ=0(Z) 0 −f ˙rp˙t f2˙t2p+ ˙r2p−2f ∂ ∂rH ℓ=0(Z) 0 i , (127) Fretrℓ=0= m0 4f h 2f′f ( ˙r2p+f ) ˙t2p H2ℓ=0(Z)+H ℓ=0(Z) 0 −f2˙rp2˙t2p−f3˙t2p+ ˙r4p+f ˙rp2 ∂ ∂rH ℓ=0(Z) 0 − ˙rp˙tp f2˙t2p+ ˙rp2+2f ∂ ∂tH ℓ=0(Z) 0 i , (128) where H2ℓ=0(Z)= 8πm0E 1 rfY 00⋆ (0, 0)H(r − rp) , (129) H0ℓ=0(Z)=8πm0E " 1 rf − 1 rpf (rp)− 1 rpf (rp)3( ˙rp) 2 # Y00⋆(0, 0)H(r − rp) . (130)
As noted in [13], the Z gauge leads to a pathological behaviour of Fαℓ=0
self approaching the horizon, see Fig. (8). It is however possible to define another gauge condition.
2. The R gauge
We thus made an other gauge choice, baptised as R. The two degrees of gauge freedom M0and M1may be chosen such that H0ℓ=0(R) = H
ℓ=0(R)
1 = H
ℓ=0(R)
2 = Hℓ=0(R) and Kℓ=0(R) = 0. The obtained monopole solution for the retarded force and the self-acceleration (SA) are now compliant with the behaviour of ℓ ≥ 2 modes, see Figs. (8, 14).
Fretrℓ=0= − m0 2f f ˙tp+ ˙rp h ˙tp f ˙rp˙tp+ ˙r2p+ 2f ∂ ∂tH ℓ=0(R) + f ˙rp2˙tp− f2˙tp+ ˙r3p+ f ˙rp ∂ ∂rH ℓ=0(R) − f′˙t p f ˙rp˙tp+ ˙rp2+ 2f Hℓ=0(R)i, (132) with Hℓ=0(R)= 8πm 0E 1 rfY ∗00(0, 0)H(r − r p(t)) . (133)
III. NUMERICAL APPROACH, PERFORMANCE AND CODE VALIDATION
A. Computation of the perturbations, and the gravitational SF
We can test the robustness and validity of our code by comparing the numerical results to the outcomes of Eqs. (104-106,107-112), knowing the analytic asymptotic behaviour for large ℓ. For the evaluation of the fields on the worldline, we use an interpolation method described in App. B.
Figure (5) shows the quantities (the wave-function and its derivatives up to third order, Hℓ
1and H2ℓ and their first derivatives) that the code is able to extract at the position of the particle during its fall from an initial rest position at r0/2M = 20. Each quantity is given for 2 ≤ ℓ ≤ 20. We plot in black the asymptotic behaviour given by Eqs. (104-106,107-112). The dashed curves are related to the side r → r−
p (superscript ”-”) and the solid curves to r → r+p (superscript ”+”). The values are in SI units of 2M/m0κ−1.
In Fig. (6), we check the asymptotic behaviour of the modes with ℓ. We observe example, at fixed rp, H1,2ℓ→∞∝ κL−1. The straight line formed by the points log10|H1,2| in terms of L has a slope −1.
Figure (7) displays the perturbation functions of the retarded field for a fall from r0/2M = 15 for 2 ≤ ℓ ≤ 20. For the modes ℓ > 8, the behaviour tends to Hℓ→∞
1,2 as expressed by Eqs. (107,110). The divergent feature of the series is due to the infinite sum of finite contributions. The standard theorem by Courant [58] states that on the 2-sphere of constant t and r, a function must be at least C2for the uniform and absolute convergence of its expansion in spherical harmonics. This condition is clearly not satisfied in the case of the radial perturbation tensor which is C0.
For the computation of the retarded force mode by mode, we use Eq. (118). The latter may provide also Fαℓ ret±, where the ± sign indicates one of the particle worldline sides. Since the ℓ-modes of the retarded field hretℓ
αβ are continuous at the position of the particle in the RW gauge, their derivatives have a jump and that is why the sign ± is needed. Indeed, the value of Fαℓ
ret±(t, rp) depends on the direction in which the derivatives are taken through the limit r → rp(t). In the following, we will consider the average of each mode only
Fretαℓ= 1 2 Fret+αℓ + Fret−αℓ . (134)
Figure (8) shows the eight first modes of the retarded force both for r and t components as a function of the particle position rp(t) for a fall from r0/2M = 15. For the ℓ = 0 mode, two curves are plotted for the Z and R gauges. The former shows a divergent behaviour as expected. The black solid line refers to the parameter Bαwhich describes the asymptotic form of Fαℓ
ret when ℓ → ∞, since because Aα+ = −Aα−, Eq. (120). The divergent feature of the series appears again due to the infinite sum of finite modal contributions. The modes tend to Bα when ℓ → ∞.
The value of the SF does not depend on the sign ”±” shown in Eq. 118). Thus, the average Fαℓ
ret± is taken for regularisation. Given Eq. (134), and considering the regularisation parameters obtained, Eq. (120), we have
Fselfα = ∞ X ℓ=0 h Fretαℓ− Bα i , (135)
2 3 0 5 10 15 20 0.0e+00 5.0e-02 1.0e-01 1.5e-01 2.0e-01 ψ + , ψ − 0 5 10 15 20 -3.0e-03 -2.0e-03 -1.0e-03 0.0e+00 1.0e-03 ∂t ψ + , ∂ t ψ 0 5 10 15 20 -1.5e-02 -1.0e-02 -5.0e-03 0.0e+00 5.0e-03 ∂r ψ + , ∂ r ψ 0 5 10 15 20 0.0e+00 2.0e-04 4.0e-04 6.0e-04 ∂ 2 tψ + , ∂ 2 tψ 0 5 10 15 20 0.0e+002.0e-03 4.0e-03 6.0e-03 8.0e-031.0e-02 ∂ 2 rψ + , ∂ 2 rψ 0 5 10 15 20 0.0e+00 5.0e-04 1.0e-03 1.5e-03 2.0e-03 ∂rt ψ + , ∂ rt ψ − 0 5 10 15 20 -2.0e-03 -1.5e-03 -1.0e-03 -5.0e-04 0.0e+005.0e-04 1.0e-03 1.5e-03 2.0e-03 ∂ 3 tψ + , ∂ 3 tψ − 0 5 10 15 20 -4.0e-03 -3.0e-03 -2.0e-03 -1.0e-03 0.0e+001.0e-03 2.0e-03 3.0e-03 4.0e-03 ∂ 3 rψ + , ∂ 3 rψ − 0 5 10 15 20 -6.0e-04 -4.0e-04 -2.0e-04 0.0e+00 2.0e-04 4.0e-04 6.0e-04 ∂rrt ψ + , ∂ rr t ψ − 0 5 10 15 20 -2.0e-04 -1.5e-04 -1.0e-04-5.0e-05 0.0e+005.0e-05 1.0e-04 1.5e-04 2.0e-04 ∂rtt ψ + , ∂ rt t ψ − 0 5 10 15 20 0.0e+005.0e-03 1.0e-02 1.5e-02 2.0e-02 2.5e-02 3.0e-02 3.5e-02 4.0e-02 H 1 0 5 10 15 20 0.0e+005.0e-03 1.0e-02 1.5e-02 2.0e-02 2.5e-02 3.0e-02 3.5e-02 4.0e-02 H 2 0 5 10 15 20 0.0e+005.0e-03 1.0e-02 1.5e-02 2.0e-02 2.5e-02 3.0e-02 3.5e-02 4.0e-02 K 0 5 10 15 20 0.0e+001.0e-04 2.0e-04 3.0e-04 4.0e-04 5.0e-04 6.0e-04 7.0e-04 8.0e-04 9.0e-04 ∂t ¯ H 1 0 5 10 15 20 rp/2ℓ 0.0e+00 5.0e-04 1.0e-03 1.5e-03 2.0e-03 ∂t ¯ H 2 0 5 10 15 20 rp/2ℓ 0.0e+005.0e-05 1.0e-04 1.5e-04 2.0e-04 2.5e-04 3.0e-04 3.5e-04 4.0e-04 ∂t ¯ K 0 5 10 15 20 rp/2ℓ 0.0e+00 5.0e-04 1.0e-03 1.5e-03 2.0e-03 ∂r ¯ H 1 0 5 10 15 20 rp/2ℓ -8.0e-04 -6.0e-04 -4.0e-04 -2.0e-04 0.0e+00 2.0e-04 4.0e-04 ∂r ¯ H 2 0 5 10 15 20 rp/2ℓ -1.5e-02 -1.0e-02 -5.0e-03 0.0e+00 5.0e-03 ∂r ¯ K 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
FIG. 5: Radial fall of a particle at rest from r0/2M = 20. The wave-function and its derivatives up to third order, H1ℓand H2ℓand their first derivatives are shown at
the position of the particle. Each quantity is given for 2 ≤ ℓ ≤ 20 (colour palette). We plot in black the asymptotic behaviour given by Eqs. (104-106,107-112). The
2 4 2.5 5.5 10.5 20.5 −4.0 −3.5 −3.0 −2.5 −2.0 −1.5 ¯ ψ 2.5 5.5 10.5 20.5 −6.0 −5.5 −5.0 −4.5 −4.0 −3.5 ∂ℓ ¯ ψ 2.5 5.5 10.5 20.5 −5.0 −4.5 −4.0 −3.5 −3.0 −2.5 ∂r ¯ ψ 2.5 5.5 10.5 20.5 −5.0 −4.8 −4.6 −4.4 −4.2 ∂ 2 ℓ¯ ψ 2.5 5.5 10.5 20.5 −3.4 −3.2 −3.0 −2.8 −2.6 ∂ 2 r¯ ψ 2.5 5.5 10.5 20.5 −4.2 −4.0 −3.8 −3.6 −3.4 −3.2 −3.0 ∂rℓ ¯ ψ -1.01 2.5 5.5 10.5 20.5 −5.8 −5.6 −5.4 −5.2 −5.0 ∂ 3 ℓ¯ ψ -0.97 2.5 5.5 10.5 20.5 −4.0 −3.8 −3.6 −3.4 −3.2 −3.0 −2.8 −2.6 ∂ 3 r¯ ψ -1.01 2.5 5.5 10.5 20.5 −5.2 −5.0 −4.8 −4.6 −4.4 −4.2 −4.0 −3.8 ∂rrℓ ¯ ψ -1.01 2.5 5.5 10.5 20.5 −5.6 −5.4 −5.2 −5.0 −4.8 −4.6 ∂rℓℓ ¯ ψ -0.98 2.5 5.5 10.5 20.5 −3.2 −3.0 −2.8 −2.6 −2.4 −2.2 −2.0 H 1 -1.0 2.5 5.5 10.5 20.5 −2.8 −2.6 −2.4 −2.2 −2.0 −1.8 −1.6 H 2 -1.0 2.5 5.5 10.5 20.5 −2.8 −2.6 −2.4 −2.2 −2.0 −1.8 −1.6 K -1.0 2.5 5.5 10.5 20.5 −4.4 −4.2 −4.0 −3.8 −3.6 −3.4 ∂ℓ ¯ H 1 -0.99 2.5 5.5 10.5 20.5 L = +1/2 −4.6 −4.4 −4.2 −4.0 −3.8 −3.6 −3.4 ∂ℓ ¯ H 2 -1.0 2.5 5.5 10.5 20.5 L = +1/2 −4.8 −4.6 −4.4 −4.2 −4.0 −3.8 ∂ℓ ¯ K -1.0 2.5 5.5 10.5 20.5 L = +1/2 −4.4 −4.2 −4.0 −3.8 −3.6 −3.4 −3.2 ∂r ¯ H 1 -1.0 2.5 5.5 10.5 20.5 L = +1/2 −4.2 −4.0 −3.8 −3.6 −3.4 −3.2 ∂r ¯ H 2 -1.0 2.5 5.5 10.5 20.5 L = +1/2 −4.2 −4.0 −3.8 −3.6 −3.4 −3.2 −3.0 ∂r ¯ K -1.0
FIG. 6: Radial fall of a particle at rest from r0/2M = 20. The quantities displayed in Fig. (5) are shown vis ´a visthe L mode for rp/2M ≈ 10. On the vertical axis,
the log10of the averaged quantities in 2M/m0κ−1 units; the average of ψℓ± provides ψ
ℓ (rp) = 1/2 ψℓ+(r p) + ψℓ−(rp)
. The slope of each straight line corresponds to
1 3 5 7 9 11 13 15 rp/2M 0 2 4 6 8 10 12 14 f( rp ) 2 + 1ℓ 1(r p ) →∞ 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 3 5 7 9 11 13 15 rp/2M 0 2 4 6 8 10 12 14 f( rp ) 2 + 1ℓ 2(r p ) →∞ 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 2 3 4 5 6 7 8 9 10 11 12 13 14 15
FIG. 7: Radial fall of a particle at rest from r0/2M = 15. The perturbation functions H1ℓ(upper panel) and H2ℓ(lower panel)
are computed at the particle position rp(t) for the modes 2 ≤ ℓ ≤ 20 (colour palette). The asymptotic behaviour of H1ℓ→∞and
Hℓ→∞
2 , Eqs. (107,110) traced in black, is confirmed.
We fix a value ℓ = ℓmax corresponding to an acceptable threshold error. The contribution of higher modes than ℓmax is computed analytically using the asymptotic behaviour with respect to L, Eqs. (137,138)
Fselfα = Fselfαℓ=0+ ℓXmax ℓ=2 Fselfαℓ | {z } numerical + ∞ X ℓ=ℓmax+1 Fselfαℓ→∞ | {z } analytic . (136)
Figure (10) shows the SF computed from the modes of the retarded force plotted in Fig. (8), for ℓmax= 8. The case corresponds to r0/2M = 15 but the general behaviour of the components of the SF remains the same regardless the value of r0. Indeed, the radial component is always oriented toward the black hole which suggests a positive work of the force during the fall (attractive nature) and therefore the energy E parameter increases [26].
Figure (10) can be compared to Fig. (3) of [13]: qualitatively the behaviour Fα
self is consistent but unlike [13] our curves do not suffer of the non-physical oscillations that pollute the first stages of the fall. We use a symmetric trajectory (m is thrown up vertically) to overcome this problem, which makes the first stage of the fall exploitable for our analysis, Sect. III B.
1 3 5 7 9 11 13 15
r
p/
2M
−0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 (4M
2/m
2 0)F
r rℓt(r
p) =0 →∞ 1.8 2.0 2.2 2.4 2.6 2.8 −0.16 −0.14 −0.12 −0.10 −0.08 −0.06 2 3 4 5 6 7 8 1 3 5 7 9 11 13 15r
p/
2M
−0.04 −0.03 −0.02 −0.01 0.00 0.01 0.02 0.03 0.04 − (4M
2/m
2 0)F
t rℓt(r
p) =0 →∞ 1.8 2.0 2.2 2.4 2.6 2.8 −0.16 −0.14 −0.12 −0.10 −0.08 −0.06 2 3 4 5 6 7 8FIG. 8: Radial fall of a particle at rest from r0/2M = 15. The average retarded force Fretαℓis computed at the particle position
rp(t) for the modes 2 ≤ ℓ ≤ 20 (colour palette). The asymptotic behaviour of Fretαℓ→∞= Bα, Eq. (120) traced in black, is
confirmed. The mode ℓ = 0 for the Z and R gauges is shown by the dot-dash and dash lines, respectively. For the former, we note the divergent behaviour at the horizon.
2.5 3.5 4.5 5.5 6.5 7.5 8.5
L
=+1/
2 −4.2 −4.0 −3.8 −3.6 −3.4 −3.2 −3.0 −2.8 lo g10 |( 4M
2/m
0 )a
sℓlf| reference (slope=-2)FIG. 9: Speed of the convergence of the series aself=Pℓa ℓ ret−Ba.
1 3 5 7 9 11 13 15
r
p/
2M
−0.15 −0.10 −0.05 0.00 0.05 − (4M
2/m
2 0)F
t selfℓ
=2 1 3 5 7 9 11 13 15r
p/
2M
−0.15 −0.10 −0.05 0.00 0.05 (4M
2/m
2 0)F
r selfℓ
=2FIG. 10: Radial fall of a particle at rest from r0/2M = 15. After the regularisation, Fig. (8), the modes are summed together
with the ℓ = 0 mode and the analytic contribution of the ℓ > 8 modes. The curves in black represent the SF, Eq. (136), for the time Ft
self(upper curve) and radial Fselfr (lower curve) components. For comparison, the quadrupole mode is also traced (red
curve), thereby showing the relevance of the modes ℓ > 2 for the SF computation.
O L−2 contained in Eq. (119). This term is computed by following the procedure in Sect. II B but keeping the higher order of development of the Green function. We obtain
Fselftℓ→∞= 15 16 m2 0E r2 pf (rp) ◦ rp 2rp ◦◦ rp− ◦ r2p L−2+ O L−4 , (137) Fselfrℓ→∞= − 15 16 m2 0E2 r2 p E2+4M rp − 1 L−2+ O L−4 , (138)
where ‘◦’ is a full derivation operator with respect to ccordinate time. The derivation of Eqs. (137,138) is obtained with a similar computation appeared in Sect. II B, but for a higher order. This is the first independent confirmation of Eqs. (6a,6b) in [13], for which derivation the reader was reminded to an accompanying paper, that finally was never published.
B. Initial conditions
The Brill-Lindquist [59] initial conditions generate quasi-normal modes and induce non-physical oscillations that pollute the first stage as shown in Figs. (11a,11c) and in [13]. We circumvent the nuisance by adopting a symmetric trajectory (m0is thrown up vertically) and consider only the portion for whichr◦p≤ 0, Figs. (11b,11d).
C. Sensitivity toℓmax
The truncation of the series in Eq. (171) depends on ℓmax, that is the highest mode to be computed numerically. Obviously, the larger is ℓmax, and more aself tends to its exact value, Fig. (12). However, to avoid the burden of an heavy numerical computation or conversely a large error on aself, we pick ℓmax such that its contribution to the truncated series is less than 0.1%. This contribution is quantified by the term ∆kaselfkℓLmax1 corresponding to the
(a) 1 3 5 7 9 R/2M −0.025 −0.020 −0.015 −0.010 −0.005 0.000 0.005 (4 M 2/m 0 ) a sℓlf
a
sℓlfa
rℓt =2 =3 =4 (b) 1 3 5 7 9 R/2M −0.025 −0.020 −0.015 −0.010 −0.005 0.000 0.005 (4 M 2/m 0 ) a sℓlfa
sℓlfa
rℓt =2 =3 =4 (c) 1 3 5 7 9 11 13 15 R/2M −0.025 −0.020 −0.015 −0.010 −0.005 0.000 0.005 (4 M 2/m 0 ) a sℓlfa
sℓlfa
rℓt =2 =3 =4 (d) 1 3 5 7 9 11 13 15 R/2M −0.025 −0.020 −0.015 −0.010 −0.005 0.000 0.005 (4 M 2/m 0 ) a sℓlfa
sℓlfa
rℓt =2 =3 =4FIG. 11: The Brill-Lindquist [59] initial conditions imposed at r = r0induce non-physical oscillations polluting the first stage
of the fall. This corresponds to the graphs (a) and (c), respectively associated to a fall from r0/2M = 10 and r0/2M = 15.
In order to exploit the first stage of the fall, we use symmetric trajectories (m is thrown up vertically) and then consider only the portion of the data corresponding to the fall into the black hole (the non-physical oscillations of the upward part vanish at infinity). This scenario is displayed in the graphs (b) and (d), respectively associated to a fall from r0/2M = 10 and
r0/2M = 15. 1 3 5 7 9 11 13 15
R/
2M
−0.015 −0.010 −0.005 0.000 0.005 0.010 0.015 (4M
2/m
0 )a
self maℓ =2 maℓ =3 maℓ =4 maℓ =5 maℓ =6 maℓ =7 maℓ =8 2.5 3.0 3.5 4.0 4.5 5.0 5.5 −0.0034 −0.0032 −0.0030 −0.0028 −0.0026 −0.0024∆kaselfkℓLmax1 =
kaselfkℓLmax1 −1− kaselfkℓLmax1
kaselfkℓLmax1
, (139)
where the L1-norm is given by the integral through the whole history of the particle position R on the background metric
kaselfkℓLmax1 =
Z
|aself|dR . (140)
Table IV gives value of aself with respect to the truncation parameter ℓmax. kaselfkℓLmax1 converges toward a finite
value such that the criterion
∆kaselfkℓLmax1 ≤ 0.1% (141)
is satisfied for ℓmax= 8.
ℓmax kaselfkℓLmax1 ∆kaselfkℓLmax1
2 0.03248 − 3 0.03042 6.8% 4 0.02960 2.7% 5 0.02922 1.3% 6 0.02904 0.6% 7 0.02896 0.2% 8 0.02893 0.1%
TABLE IV: Estimate of ℓmax for a given accuracy.
D. Sensitivity toh
The grid step parameter h must be chosen carefully to reach the desired accuracy without useless extra computation. We follow the same reasoning with ℓmax considering
∆kaselfkhLk1 = kaselfkhLk−11 − kaselfk hk L1 kaselfkhLk1 , (142)
where kaselfkhLk1 corresponds to self-acceleration computed with ℓmax = 8 with an integration step h/2M = hk such
that h0= 0.01, h1= 0.005, h2= 0.0025, h3= 0.001. It is found that the criterion
∆kaselfkhL1 ≤ 0.1% (143)
E. Asymptoticℓ-behaviour
In section II, we observed that the code assured the correct asymptotic behaviour of quantities for large ℓ, Fig. (6). We confirm that the regularisation technique works and the regularisation parameters are computed correctly, through the asymptotic behaviour of the self-quantities (acceleration and force), with respect to L.
Figure (9) exhibits the values of aℓ
self in terms of L. The slope of the line indicates the rate of the convergence of the series aself=Pℓaℓret− Baα, that is O L−2
. We recall that aℓ
ret is an average aℓret = 1/2(aℓret++ aℓret-). A good behaviour has been found for several values of R ∈ [2M, r0] and of r0≥ 10. Figure (13) also displays a good exhibit of values for the two components of the SF.
2.5 3.5 4.5 5.5 6.5 7.5 8.5
L
=+1/
2 −4.5 −4.0 −3.5 −3.0 lo g10 |( 4M
2/m
2 0)F
α sℓlf|ref. for t component ref. for r component
FIG. 13: Speed of convergence of the series Fα self =
P
ℓF αℓ
ret−Bα for both components of the SF t (continuous line) and r
(dashed line).
IV. EQUATIONS OF MOTION
The geodesic equation of motion of a test particle in the SD spacetime is uα∇
βuβ= 0 . (144)
In radial fall, the angular momentum is zero (L = 0), and without loss of generality, the azimuthal angle is chosen to be null too (θ = 0). In coordinate time the geodesic equation is given by [61]
d2r p dt2 = a0(rp, ◦ rp) = − Γrαβ− ◦ rpΓtαβ ◦ xαp ◦ xβp = − 1 2f (rp)f ′(r p) " 1 −f (r3 p)2 drp dt 2# = f (rp)f′(rp) 1 −32f (rp) E2 . (145) Instead, the perturbed motion is seen as an accelerated motion in the background SD spacetime. The worldline xα
p(τ ) is not geodesic anymore. The equation of motion changes into
m0uα∇βuβ= Fselfα , (146)
where Fα
self∼ O (m0/M ) is the SF computed in the RW gauge. Equation (146) can be rewritten in coordinate time
d2r p dt2 = a0(rp, ◦ rp) + aself(rp, ◦ rp) . (147)
A. Pragmatic approach
In the pragmatic approach [25, 26, 60, 62], we consider Eq. (147) in its linearised version at first order around the reference geodesic Xα(τ ) which is the solution of
d2Xα dτ2 + Γ α βγ(Xα) dXβ dτ dXγ dτ = 0 , (149) where Γα
βγ(Xα) is the affine connection associated to the background metric. The perturbed trajectory labelled by the coordinates xαp(τ ) = (t, rp) is the solution of Eq. (146), developed as
d2xα p dτ2 + Γ α βγ(xαp) dxβ p dτ dxγ p dτ = Fα self m0 . (150)
It differs from the reference geodesic by ∆Xα∝ m
0/M such that xαp = Xα+ ∆Xα, (151) ˙xαp = ˙Xα+ ∆ ˙Xα, (152) ¨ xα p = ¨Xα+ ∆ ¨Xα, (153)
having supposed that the perturbed motion remains close to the geodesic. By injecting Eqs. (151-153) into Eq. (150), we have d2 dτ2 Xα+ ∆Xα+Γαβγ+ ∂δΓαβγ∆Xδ d dτ Xβ+ ∆Xβ d dτ Xγ+ ∆Xγ=F α self m0 . (154) For d/dτ = (dt/dτ )d/dt, and d2/dτ2 = (d2t/dτ2)d/dt + (dt/dτ )2
d2/dt2, Eq. (154) is expanded to first order in coordinate time d2t dτ2 dτ dt 2 (X◦α+ ∆X◦α) +X◦◦α+ ∆X◦◦α+ Γα βγ ◦ XβX◦γ+ 2X◦γΓα βγ ◦ Xβ+ ∆Xδ∂ δΓαβγ ◦ XβX◦γ =Fselfα m0 dτ dt 2 . (155) Assuming ∆Xα= σα∆R with σα
··= (0, 1) we find for the time and radial components d2t dτ2 dτ dt 2 = F t self m0 d2τ dt2 2 − Γtβγ ◦ XβX◦γ− 2Γtβγ ◦ Xβσγ∆R − ∂◦ rΓtβγ ◦ XβX◦γ∆R , (156) d2t dτ2 dτ dt 2 d dt R + ∆R+R + ∆◦◦ R + Γ◦◦ rβγ ◦ XβX◦γ+ 2Γrβγ ◦ Xβσγ∆R + ∂◦ rΓrβγ ◦ XβX◦γ∆R = F r self m0 dτ dt 2 . (157)
By injecting Eq. (156) into Eq. (157), we find at first order
◦◦ R + ∆R +◦◦ Γr βγ− Γtβγ ◦ RX◦βX◦γ+ ∆R∂ rΓrβγ− ∂rΓtβγ ◦ RX◦βX◦γ+ ∆R◦2Γrβγσγ− 2Γtβγσγ ◦ R − Γtβγ ◦ XγX◦β= ˙t 2 m0 Fselfr (R, ◦ R) −RF◦ selft (R, ◦ R) , (158)
